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Suppose the regression model is $\text{wage} = \beta_1x_1 + u$. The conditional expectation function is $\operatorname{E}[\text{wage} \mid x_1] = \beta_1x_1$. Then, we interpet $\beta_1$ as the change in $\text{wage}$ if $x_1$ changes by one unit "on average" because what is changing is the expected value of wage. Now consider the model $\text{lwage} = \beta_1x_1 + u$. The conditional expectation function is $\operatorname{E}[\text{lwage} \mid x_1] = \beta_1x_1$. If I do not take the expectation, it is straightforward to see that $\beta_1$ gives the percentage change in wage when $x_1$ changes by one unit. But I wonder what the change in the expected value of log entails. I somehow want to end up with the "on average" interpretation.

Michael Hardy
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Snoopy
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  • Possible duplicate of [Interpretation of log transformed predictor](https://stats.stackexchange.com/questions/18480/interpretation-of-log-transformed-predictor) – COOLSerdash Dec 05 '18 at 13:33
  • I am dealing with the expected value in this question. Is it really duplicate? – Snoopy Dec 05 '18 at 13:35
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    Yes, the regression models the conditional expectation of the dependent variable. – COOLSerdash Dec 05 '18 at 13:37
  • Yes of course but the linked thread is not explicit about expectations. Why is my question a possible duplicate? – Snoopy Dec 05 '18 at 13:40
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    The linked thread just doesn't explicitly spell it out, but it is implied. If you just add the word "on average", you have your answer (unless I completely misunderstood the question). So for the log-transformed wage, the interpretation for $\beta_1$ is: "For each unit increase in $x_1$, wages changes by $\beta_1\times 100$ percent on average". – COOLSerdash Dec 05 '18 at 13:44
  • But I do not want to keep things implied. That is my question. And I am not sure if your answer is correct because expected value of log is not log of expected value and I am suspecting that this will be part of the answer. I do not agree that this is a possible duplicate. If it is, I will delete my question. – Snoopy Dec 05 '18 at 13:46
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    I am not sure what the answer s here but it might be good to edit your question with why you do not see it as a duplicate so people can argue for or against that. – mdewey Dec 05 '18 at 15:04

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